Sociable number

In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918.^[1] In a sociable sequence, each number is the sum of the proper divisors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.

The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.

If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up to ${\displaystyle 5\times 10^{7}}$ as of 1970.^[2]

It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.

Example

As an example, the number 1,264,460 is a sociable number whose cyclic aliquot sequence has a period of 4:

The sum of the proper divisors of ${\displaystyle 1264460}$  (${\displaystyle =2^{2}\cdot 5\cdot 17\cdot 3719}$ ) is

1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860,

the sum of the proper divisors of ${\displaystyle 1547860}$  (${\displaystyle =2^{2}\cdot 5\cdot 193\cdot 401}$ ) is

1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636,

the sum of the proper divisors of ${\displaystyle 1727636}$  (${\displaystyle =2^{2}\cdot 521\cdot 829}$ ) is

1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184, and

the sum of the proper divisors of ${\displaystyle 1305184}$  (${\displaystyle =2^{5}\cdot 40787}$ ) is

1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.

List of known sociable numbers

The following categorizes all known sociable numbers as of July 2018^[update] by the length of the corresponding aliquot sequence:

Sequence length	Number of known sequences	lowest number in sequence^[3]
1 (Perfect number)	51	6
2 (Amicable number)	1225736919^[4]	220
4	5398	1,264,460
5	1	12,496
6	5	21,548,919,483
8	4	1,095,447,416
9	1	805,984,760
28	1	14,316

It is conjectured that if n is congruent to 3 modulo 4 then there is no such sequence with length n.

The 5-cycle sequence is: 12496, 14288, 15472, 14536, 14264

The only known 28-cycle is: 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716 (sequence A072890 in the OEIS). It was discovered by Ben Orlin.

These two sequences provide the only sociable numbers below 1 million (other than the perfect and amicable numbers).

Searching for sociable numbers

The aliquot sequence can be represented as a directed graph, ${\displaystyle G_{n,s}}$ , for a given integer ${\displaystyle n}$ , where ${\displaystyle s(k)}$  denotes the sum of the proper divisors of ${\displaystyle k}$ .^[5]Cycles in ${\displaystyle G_{n,s}}$  represent sociable numbers within the interval ${\displaystyle [1,n]}$ . Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

Conjecture of the sum of sociable number cycles

It is conjectured that as the number of sociable number cycles with length greater than 2 approaches infinity, the proportion of the sums of the sociable number cycles divisible by 10 approaches 1 (sequence A292217 in the OEIS).

References

1. P. Poulet, #4865, L'Intermédiaire des Mathématiciens 25 (1918), pp. 100–101. (The full text can be found at ProofWiki: Catalan-Dickson Conjecture.)
2. Bratley, Paul; Lunnon, Fred; McKay, John (1970). "Amicable numbers and their distribution". Mathematics of Computation. 24 (110): 431–432. doi:10.1090/S0025-5718-1970-0271005-8. ISSN 0025-5718.
3. https://oeis.org/A003416 cross referenced with https://oeis.org/A052470
4. Sergei Chernykh Amicable pairs list
5. Rocha, Rodrigo Caetano; Thatte, Bhalchandra (2015), Distributed cycle detection in large-scale sparse graphs, Simpósio Brasileiro de Pesquisa Operacional (SBPO), doi:10.13140/RG.2.1.1233.8640

• H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423–429

External links

• A list of known sociable numbers
• Weisstein, Eric W. "Sociable numbers". MathWorld.
• A003416 (smallest sociable number from each cycle) and A122726 (all sociable numbers) in OEIS